Question: In this problem we have the symmetric matrix

\[A=\left(\begin{array}{ccc} 2 & 0 & 0 \\ 0 & 3 & -4 \\ 0 & -4 & 3 \end{array}\right)\]

1. Calculate the characteristic polynomial \(\operatorname{det}\left(A-\lambda I_{3}\right)\).
2. Find the eigenvalues of \(A\).
3. For each eigenvalue of \(A\), find an orthonormal basis| for the corresponding eigenspace.
4. Find an orthogonal matrix \(P\) and a diagonal matrix \(D\) so that \(P^{t} A P=D\).
5. Show that for each real number \(\mu\) and for each \(n \geq 1,\left(A+\mu I_{3}\right)^{n}=P\left(D+\mu I_{3}\right)^{n} P^{t} .\) In particular, calculate \(\left(A+I_{3}\right)^{3}\).

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